Propagating Absoluteness Downwards

We assume that, in a universe of set theory, some projective level of absoluteness holds for the collapse forcing, i.e., the forcing notion collapsing some cardinal to be countable. It is well-known that every other forcing notion of no greater size embeds into the collapse. Hence a natural  question is whether our initial assumption suffices to achieve the same level of projective absoluteness for those smaller forcing notions. We will give an answer to this question under suitable large cardinal assumptions, following a path that will lead us through determinacy axioms, inner model theory, homogeneity measures, and universally Baire sets.